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We know that two quantum states, can not get entangled via local classical interactions/communication (LOCC). However, do two quantum states locally interacting via quantum interactions always get entangled?

Suppose, two Quantum systems, initially in the unentangled state, interact locally with a mediator, and no entanglement is generated between the two because of interaction, can we unambiguously determine that the mediator was classical in nature?

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  • $\begingroup$ what is a “mediator” in this case? $\endgroup$ Jun 30, 2022 at 11:44
  • $\begingroup$ Say, the agent that locally interacts with the two systems and transmits information, like a field for example. $\endgroup$
    – Paranoid
    Jun 30, 2022 at 12:44
  • $\begingroup$ what does "locally interacting via quantum interactions" mean precisely here? Are you referring to an interaction via a specific type of Hamiltonian (eg one with pairwise interactions)? $\endgroup$
    – glS
    Jul 2, 2022 at 8:12

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Answer

However, do two quantum states locally interacting via quantum interactions always get entangled?

No. I can give you an example using the $CNOT$ gate for qubits which is defined by the following relation:

$$ CNOT |0\rangle |0\rangle = |0 \rangle |0\rangle $$ $$ CNOT |0\rangle |1\rangle = |0 \rangle |1\rangle $$ $$ CNOT |1\rangle |0\rangle = |1 \rangle |1\rangle $$ $$ CNOT |1\rangle |1\rangle = |1 \rangle |0\rangle $$

i.e. the second qubit is flipped if the first one is in state $|1\rangle$. Clearly none of the above states is entangled after operating with CNOT.This is because the state can be written in the form $|\cdot\rangle_{q_1}|\cdot\rangle_{q_2}$ where $q_1$ and $q_2$ are qubits 1 and 2.

Now, look at the not-entangled state $$\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)\otimes |1\rangle = \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle|1\rangle)$$

after acting with $CNOT$ we obtain: $$CNOT \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle|1\rangle) = \frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle |0\rangle)$$

using the defining relations of the CNOT gate. The state $\frac{1}{\sqrt 2} (|0\rangle|1\rangle + |1\rangle |0\rangle)$ is called Bell state and it is a maximally entangled state. Hence, you can see that the final level of entanglement also depends on the initial state that experiences the interaction.

Extension: Entangling Power

After doing some more reading, I came across the concept of entangling power. For a gate $U$, the entangling power $K$ is defined as [1]:

$$K_{E}(U)=\max _{|\phi\rangle,|\psi\rangle} E(U(|\phi\rangle|\psi\rangle))$$

This quantity is the maximum attainable entanglement $E$ maximised over all states $|\phi\rangle|\psi\rangle$.

[1] https://iopscience.iop.org/article/10.1088/1751-8121/aad7cb/pdf

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